A scaled difference chi-square test statistic for moment structure analysis

A family of scaling corrections aimed to improve the chi-square approximation of goodness-of-fit test statistics in small samples, large models, and nonnormal data was proposed in Satorra and Bentler (1994). For structural equations models, Satorra-Bentler's (SB) scaling corrections are available in standard computer software. Often, however, the interest is not on the overall fit of a model, but on a test of the restrictions that a null model sayM 0 implies on a less restricted oneM 1. IfT 0 andT 1 denote the goodness-of-fit test statistics associated toM 0 andM 1, respectively, then typically the differenceT d =T 0−T 1 is used as a chi-square test statistic with degrees of freedom equal to the difference on the number of independent parameters estimated under the modelsM 0 andM 1. As in the case of the goodness-of-fit test, it is of interest to scale the statisticT d in order to improve its chi-square approximation in realistic, that is, nonasymptotic and nonormal, applications. In a recent paper, Satorra (2000) shows that the difference between two SB scaled test statistics for overall model fit does not yield the correct SB scaled difference test statistic. Satorra developed an expression that permits scaling the difference test statistic, but his formula has some practical limitations, since it requires heavy computations that are not available in standard computer software. The purpose of the present paper is to provide an easy way to compute the scaled difference chi-square statistic from the scaled goodness-of-fit test statistics of modelsM 0 andM 1. A Monte Carlo study is provided to illustrate the performance of the competing statistics.